Chapter 5 Section 5.1 Angles

An angle is formed when two rays are joined at a common endpoint. The point where they join is called the vertex. The ray where the and begins is called the initial side and the ray where it ends is called the terminal side. To measure the angle is to associate a number with it in a consistent manner. The two most common ways to do that are using degrees and radians as units of measure. If the direction from the initial side to the terminal side is counterclockwise the angle measure is positive. If the direction from the initial side to the terminal side is clockwise the angle measure is negative. If the initial side of the angle is a horizontal ray pointing to the right with the vertex of the angle at the origin of an xy - coordinate plane the angle is said to be in standard position. Initial side Terminal side Positive angle measure Terminal side Initial side Negative angle measure x y

Degree Measure The degree measurement of an angle is based on the degree measure of an entire circle being 360  and any fraction of the circle will be proportional (i.e. will form the same fraction). The examples below show the measure of angles in standard position. x y ½ circle ½ · 360  = 180  x y ¼ circle ¼ · 360  = 90  x y 1/8 circle 1/8 · 360  = 45  x y 5/8 circle 5/8 · 360  = 225  x y 1 circle (neg) 1 · -360  = -360  x y 3/8 circle (neg) 3/8 · -360  = -135  x y 1/12 circle (neg) 1/12 · -360  = -30  x y ¾ circle (neg) ¾ · -360  = -270 

Find the angles :

Coterminal Angles Any two angles with the same terminal side are said to be coterminal angles. We have seen that two angles that have the same terminal side can have different measures depending if you are going clockwise (negative) or counterclockwise (positive) around the circle. Angles exceeding 360  or 2  radians are thought of a wrapping around the circle a certain number of times before hitting the terminal angle. For example a 780  angle is really coterminal with a 60  angle since: 780   = 420  and 420   = 60  x y 3/8 circle (neg) 3/8 ·-360  = -135  x y 5/8 circle (pos) 5/8 ·360  = 225  x = 13/8 circle (pos) 13/8 ·360  = 585  y